Answer :
To expand the binomial [tex]\((-3x + 7)^3\)[/tex], we can use the Binomial Theorem, which states:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k
\][/tex]
For the given problem, we have [tex]\(a = -3x\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(n = 3\)[/tex].
Let's calculate it step-by-step:
1. Identify the terms of the binomial expansion:
[tex]\[
(-3x + 7)^3 = \sum_{k=0}^{3} {3 \choose k} (-3x)^{3-k} (7)^k
\][/tex]
2. Calculate each term in the expansion:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
{3 \choose 0} (-3x)^{3-0} (7)^0 = 1 \cdot (-3x)^3 \cdot 1 = -27x^3
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
{3 \choose 1} (-3x)^{3-1} (7)^1 = 3 \cdot (-3x)^2 \cdot 7 = 3 \cdot 9x^2 \cdot 7 = 189x^2
\][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[
{3 \choose 2} (-3x)^{3-2} (7)^2 = 3 \cdot (-3x)^1 \cdot 49 = 3 \cdot (-3x) \cdot 49 = -441x
\][/tex]
- For [tex]\(k = 3\)[/tex]:
[tex]\[
{3 \choose 3} (-3x)^{3-3} (7)^3 = 1 \cdot 1 \cdot 343 = 343
\][/tex]
3. Combine all the terms:
[tex]\[
-27x^3 + 189x^2 - 441x + 343
\][/tex]
So, the binomial expansion of [tex]\((-3x + 7)^3\)[/tex] is:
[tex]\[
-27x^3 + 189x^2 - 441x + 343
\][/tex]
From the given options, the correct answer is:
d. [tex]\(-27x^3 + 189x^2 - 441x + 343\)[/tex]
[tex]\[
(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k
\][/tex]
For the given problem, we have [tex]\(a = -3x\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(n = 3\)[/tex].
Let's calculate it step-by-step:
1. Identify the terms of the binomial expansion:
[tex]\[
(-3x + 7)^3 = \sum_{k=0}^{3} {3 \choose k} (-3x)^{3-k} (7)^k
\][/tex]
2. Calculate each term in the expansion:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
{3 \choose 0} (-3x)^{3-0} (7)^0 = 1 \cdot (-3x)^3 \cdot 1 = -27x^3
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
{3 \choose 1} (-3x)^{3-1} (7)^1 = 3 \cdot (-3x)^2 \cdot 7 = 3 \cdot 9x^2 \cdot 7 = 189x^2
\][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[
{3 \choose 2} (-3x)^{3-2} (7)^2 = 3 \cdot (-3x)^1 \cdot 49 = 3 \cdot (-3x) \cdot 49 = -441x
\][/tex]
- For [tex]\(k = 3\)[/tex]:
[tex]\[
{3 \choose 3} (-3x)^{3-3} (7)^3 = 1 \cdot 1 \cdot 343 = 343
\][/tex]
3. Combine all the terms:
[tex]\[
-27x^3 + 189x^2 - 441x + 343
\][/tex]
So, the binomial expansion of [tex]\((-3x + 7)^3\)[/tex] is:
[tex]\[
-27x^3 + 189x^2 - 441x + 343
\][/tex]
From the given options, the correct answer is:
d. [tex]\(-27x^3 + 189x^2 - 441x + 343\)[/tex]
